Problem: $\sum\limits_{k=1}^{775 }{{(-3k -4)}}=$
Answer: What is the question asking for? The question is asking for the sum of the values of $-3k -4$ from $k = 1$ to $k = 775 $ : $(-3 \cdot 1 -4) + (-3 \cdot 2 -4) +... + (-3\cdot {775} -4)$ The series is arithmetic because the formula $-3k -4$ is a linear function of $k$. Formula for arithmetic series The sum $S_n$ of a finite arithmetic series is $S_n = \dfrac {\left(a_1 + a_n \right)}{2} \cdot n$ where $a_1$ is the first term, $a_n$ is the last term, and $n$ is the number of terms. What do we need to use the formula? The number of terms $(n = {775})$ is the upper limit of the sigma notation. We need to find $a_1$ (the first term) and $a_{775}$ (the last term). Step 1: Find $a_1$ and $a_{775}$ (the first and the last term) $a_1 = -3(1) -4 = {-7}$ $a_{775} = -3(775) -4 = {-2329}$ Step 2: Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac {\left(a_1 + a_n \right)}{2} \cdot n \\\\ S_{{775}}&= \dfrac {\left({-7} + ({-2329}) \right)}{2} \cdot {775} \\\\ S_{{775}} &= -1168 \left(775\right) \\\\ S_{{775}} &= -905{,}200\end{aligned}$ The answer $ -905{,}200 $